wenzelm@23252: (* Title: HOL/Groebner_Basis.thy wenzelm@23252: ID: $Id$ wenzelm@23252: Author: Amine Chaieb, TU Muenchen wenzelm@23252: *) wenzelm@23252: wenzelm@23252: header {* Semiring normalization and Groebner Bases *} haftmann@28402: wenzelm@23252: theory Groebner_Basis haftmann@28402: imports Arith_Tools wenzelm@23252: uses wenzelm@23252: "Tools/Groebner_Basis/misc.ML" wenzelm@23252: "Tools/Groebner_Basis/normalizer_data.ML" wenzelm@23252: ("Tools/Groebner_Basis/normalizer.ML") chaieb@23312: ("Tools/Groebner_Basis/groebner.ML") wenzelm@23252: begin wenzelm@23252: wenzelm@23252: subsection {* Semiring normalization *} wenzelm@23252: wenzelm@23252: setup NormalizerData.setup wenzelm@23252: wenzelm@23252: wenzelm@23258: locale gb_semiring = wenzelm@23252: fixes add mul pwr r0 r1 wenzelm@23252: assumes add_a:"(add x (add y z) = add (add x y) z)" wenzelm@23252: and add_c: "add x y = add y x" and add_0:"add r0 x = x" wenzelm@23252: and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x" wenzelm@23252: and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0" wenzelm@23252: and mul_d:"mul x (add y z) = add (mul x y) (mul x z)" wenzelm@23252: and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)" wenzelm@23252: begin wenzelm@23252: wenzelm@23252: lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)" wenzelm@23252: proof (induct p) wenzelm@23252: case 0 wenzelm@23252: then show ?case by (auto simp add: pwr_0 mul_1) wenzelm@23252: next wenzelm@23252: case Suc wenzelm@23252: from this [symmetric] show ?case wenzelm@23252: by (auto simp add: pwr_Suc mul_1 mul_a) wenzelm@23252: qed wenzelm@23252: wenzelm@23252: lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)" wenzelm@23252: proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1) wenzelm@23252: fix q x y wenzelm@23252: assume "\x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)" wenzelm@23252: have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))" wenzelm@23252: by (simp add: mul_a) wenzelm@23252: also have "\ = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c) wenzelm@23252: also have "\ = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a) wenzelm@23252: finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) = wenzelm@23252: mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c) wenzelm@23252: qed wenzelm@23252: wenzelm@23252: lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)" wenzelm@23252: proof (induct p arbitrary: q) wenzelm@23252: case 0 wenzelm@23252: show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto wenzelm@23252: next wenzelm@23252: case Suc wenzelm@23252: thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc) wenzelm@23252: qed wenzelm@23252: wenzelm@23252: wenzelm@23252: subsubsection {* Declaring the abstract theory *} wenzelm@23252: wenzelm@23252: lemma semiring_ops: ballarin@28699: fixes meta_term :: "'a => prop" ("TERM _") wenzelm@23252: shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)" wenzelm@23252: and "TERM r0" and "TERM r1" wenzelm@23252: by rule+ wenzelm@23252: wenzelm@23252: lemma semiring_rules: wenzelm@23252: "add (mul a m) (mul b m) = mul (add a b) m" wenzelm@23252: "add (mul a m) m = mul (add a r1) m" wenzelm@23252: "add m (mul a m) = mul (add a r1) m" wenzelm@23252: "add m m = mul (add r1 r1) m" wenzelm@23252: "add r0 a = a" wenzelm@23252: "add a r0 = a" wenzelm@23252: "mul a b = mul b a" wenzelm@23252: "mul (add a b) c = add (mul a c) (mul b c)" wenzelm@23252: "mul r0 a = r0" wenzelm@23252: "mul a r0 = r0" wenzelm@23252: "mul r1 a = a" wenzelm@23252: "mul a r1 = a" wenzelm@23252: "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" wenzelm@23252: "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" wenzelm@23252: "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" wenzelm@23252: "mul (mul lx ly) rx = mul (mul lx rx) ly" wenzelm@23252: "mul (mul lx ly) rx = mul lx (mul ly rx)" wenzelm@23252: "mul lx (mul rx ry) = mul (mul lx rx) ry" wenzelm@23252: "mul lx (mul rx ry) = mul rx (mul lx ry)" wenzelm@23252: "add (add a b) (add c d) = add (add a c) (add b d)" wenzelm@23252: "add (add a b) c = add a (add b c)" wenzelm@23252: "add a (add c d) = add c (add a d)" wenzelm@23252: "add (add a b) c = add (add a c) b" wenzelm@23252: "add a c = add c a" wenzelm@23252: "add a (add c d) = add (add a c) d" wenzelm@23252: "mul (pwr x p) (pwr x q) = pwr x (p + q)" wenzelm@23252: "mul x (pwr x q) = pwr x (Suc q)" wenzelm@23252: "mul (pwr x q) x = pwr x (Suc q)" wenzelm@23252: "mul x x = pwr x 2" wenzelm@23252: "pwr (mul x y) q = mul (pwr x q) (pwr y q)" wenzelm@23252: "pwr (pwr x p) q = pwr x (p * q)" wenzelm@23252: "pwr x 0 = r1" wenzelm@23252: "pwr x 1 = x" wenzelm@23252: "mul x (add y z) = add (mul x y) (mul x z)" wenzelm@23252: "pwr x (Suc q) = mul x (pwr x q)" wenzelm@23252: "pwr x (2*n) = mul (pwr x n) (pwr x n)" wenzelm@23252: "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))" wenzelm@23252: proof - wenzelm@23252: show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp wenzelm@23252: next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp wenzelm@23252: next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp wenzelm@23252: next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp wenzelm@23252: next show "add r0 a = a" using add_0 by simp wenzelm@23252: next show "add a r0 = a" using add_0 add_c by simp wenzelm@23252: next show "mul a b = mul b a" using mul_c by simp wenzelm@23252: next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp wenzelm@23252: next show "mul r0 a = r0" using mul_0 by simp wenzelm@23252: next show "mul a r0 = r0" using mul_0 mul_c by simp wenzelm@23252: next show "mul r1 a = a" using mul_1 by simp wenzelm@23252: next show "mul a r1 = a" using mul_1 mul_c by simp wenzelm@23252: next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" wenzelm@23252: using mul_c mul_a by simp wenzelm@23252: next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" wenzelm@23252: using mul_a by simp wenzelm@23252: next wenzelm@23252: have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c) wenzelm@23252: also have "\ = mul rx (mul ry (mul lx ly))" using mul_a by simp wenzelm@23252: finally wenzelm@23252: show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" wenzelm@23252: using mul_c by simp wenzelm@23252: next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp wenzelm@23252: next wenzelm@23252: show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a) wenzelm@23252: next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a ) wenzelm@23252: next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c) wenzelm@23252: next show "add (add a b) (add c d) = add (add a c) (add b d)" wenzelm@23252: using add_c add_a by simp wenzelm@23252: next show "add (add a b) c = add a (add b c)" using add_a by simp wenzelm@23252: next show "add a (add c d) = add c (add a d)" wenzelm@23252: apply (simp add: add_a) by (simp only: add_c) wenzelm@23252: next show "add (add a b) c = add (add a c) b" using add_a add_c by simp wenzelm@23252: next show "add a c = add c a" by (rule add_c) wenzelm@23252: next show "add a (add c d) = add (add a c) d" using add_a by simp wenzelm@23252: next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr) wenzelm@23252: next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp wenzelm@23252: next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp wenzelm@23252: next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c) wenzelm@23252: next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul) wenzelm@23252: next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr) wenzelm@23252: next show "pwr x 0 = r1" using pwr_0 . wenzelm@23252: next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c) wenzelm@23252: next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp wenzelm@23252: next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp wenzelm@23252: next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr) wenzelm@23252: next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))" wenzelm@23252: by (simp add: nat_number pwr_Suc mul_pwr) wenzelm@23252: qed wenzelm@23252: wenzelm@23252: wenzelm@26462: lemmas gb_semiring_axioms' = wenzelm@26314: gb_semiring_axioms [normalizer wenzelm@23252: semiring ops: semiring_ops wenzelm@26314: semiring rules: semiring_rules] wenzelm@23252: wenzelm@23252: end wenzelm@23252: wenzelm@23258: interpretation class_semiring: gb_semiring wenzelm@23252: ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"] nipkow@23477: by unfold_locales (auto simp add: ring_simps power_Suc) wenzelm@23252: wenzelm@23252: lemmas nat_arith = wenzelm@23252: add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of wenzelm@23252: wenzelm@23252: lemma not_iszero_Numeral1: "\ iszero (Numeral1::'a::number_ring)" wenzelm@23252: by (simp add: numeral_1_eq_1) wenzelm@23252: lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False wenzelm@23252: if_True add_0 add_Suc add_number_of_left mult_number_of_left wenzelm@23252: numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1 wenzelm@23252: numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1 huffman@26086: iszero_number_of_Bit1 iszero_number_of_Bit0 nonzero_number_of_Min wenzelm@23252: iszero_number_of_Pls iszero_0 not_iszero_Numeral1 wenzelm@23252: wenzelm@23252: lemmas semiring_norm = comp_arith wenzelm@23252: wenzelm@23252: ML {* wenzelm@23573: local wenzelm@23252: wenzelm@23573: open Conv; wenzelm@23573: wenzelm@23573: fun numeral_is_const ct = wenzelm@23573: can HOLogic.dest_number (Thm.term_of ct); wenzelm@23252: wenzelm@23573: fun int_of_rat x = wenzelm@23573: (case Rat.quotient_of_rat x of (i, 1) => i wenzelm@23573: | _ => error "int_of_rat: bad int"); wenzelm@23252: wenzelm@23573: val numeral_conv = wenzelm@23573: Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv wenzelm@23573: Simplifier.rewrite (HOL_basic_ss addsimps wenzelm@23573: (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)})); wenzelm@23573: wenzelm@23573: in wenzelm@23573: wenzelm@23573: fun normalizer_funs key = wenzelm@23573: NormalizerData.funs key wenzelm@23252: {is_const = fn phi => numeral_is_const, wenzelm@23252: dest_const = fn phi => fn ct => wenzelm@23252: Rat.rat_of_int (snd wenzelm@23252: (HOLogic.dest_number (Thm.term_of ct) wenzelm@23252: handle TERM _ => error "ring_dest_const")), wenzelm@23573: mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x), chaieb@23330: conv = fn phi => K numeral_conv} wenzelm@23573: wenzelm@23573: end wenzelm@23252: *} wenzelm@23252: wenzelm@26462: declaration {* normalizer_funs @{thm class_semiring.gb_semiring_axioms'} *} wenzelm@23573: wenzelm@23252: wenzelm@23258: locale gb_ring = gb_semiring + wenzelm@23252: fixes sub :: "'a \ 'a \ 'a" wenzelm@23252: and neg :: "'a \ 'a" wenzelm@23252: assumes neg_mul: "neg x = mul (neg r1) x" wenzelm@23252: and sub_add: "sub x y = add x (neg y)" wenzelm@23252: begin wenzelm@23252: wenzelm@23252: lemma ring_ops: ballarin@28699: fixes meta_term :: "'a => prop" ("TERM _") wenzelm@23252: shows "TERM (sub x y)" and "TERM (neg x)" . wenzelm@23252: wenzelm@23252: lemmas ring_rules = neg_mul sub_add wenzelm@23252: wenzelm@26462: lemmas gb_ring_axioms' = wenzelm@26314: gb_ring_axioms [normalizer wenzelm@26314: semiring ops: semiring_ops wenzelm@26314: semiring rules: semiring_rules wenzelm@26314: ring ops: ring_ops wenzelm@26314: ring rules: ring_rules] wenzelm@23252: wenzelm@23252: end wenzelm@23252: wenzelm@23252: wenzelm@23258: interpretation class_ring: gb_ring ["op +" "op *" "op ^" wenzelm@23252: "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"] wenzelm@23252: by unfold_locales simp_all wenzelm@23252: wenzelm@23252: wenzelm@26462: declaration {* normalizer_funs @{thm class_ring.gb_ring_axioms'} *} wenzelm@23252: wenzelm@23252: use "Tools/Groebner_Basis/normalizer.ML" wenzelm@23252: chaieb@27666: wenzelm@23252: method_setup sring_norm = {* wenzelm@23252: Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt)) wenzelm@23458: *} "semiring normalizer" wenzelm@23252: wenzelm@23252: chaieb@23327: locale gb_field = gb_ring + chaieb@23327: fixes divide :: "'a \ 'a \ 'a" chaieb@23327: and inverse:: "'a \ 'a" chaieb@23327: assumes divide: "divide x y = mul x (inverse y)" chaieb@23327: and inverse: "inverse x = divide r1 x" chaieb@23327: begin chaieb@23327: wenzelm@26462: lemmas gb_field_axioms' = wenzelm@26314: gb_field_axioms [normalizer wenzelm@26314: semiring ops: semiring_ops wenzelm@26314: semiring rules: semiring_rules wenzelm@26314: ring ops: ring_ops wenzelm@26314: ring rules: ring_rules] chaieb@23327: chaieb@23327: end chaieb@23327: wenzelm@23458: wenzelm@23266: subsection {* Groebner Bases *} wenzelm@23252: wenzelm@23258: locale semiringb = gb_semiring + wenzelm@23252: assumes add_cancel: "add (x::'a) y = add x z \ y = z" wenzelm@23252: and add_mul_solve: "add (mul w y) (mul x z) = wenzelm@23252: add (mul w z) (mul x y) \ w = x \ y = z" wenzelm@23252: begin wenzelm@23252: wenzelm@23252: lemma noteq_reduce: "a \ b \ c \ d \ add (mul a c) (mul b d) \ add (mul a d) (mul b c)" wenzelm@23252: proof- wenzelm@23252: have "a \ b \ c \ d \ \ (a = b \ c = d)" by simp wenzelm@23252: also have "\ \ add (mul a c) (mul b d) \ add (mul a d) (mul b c)" wenzelm@23252: using add_mul_solve by blast wenzelm@23252: finally show "a \ b \ c \ d \ add (mul a c) (mul b d) \ add (mul a d) (mul b c)" wenzelm@23252: by simp wenzelm@23252: qed wenzelm@23252: wenzelm@23252: lemma add_scale_eq_noteq: "\r \ r0 ; (a = b) \ ~(c = d)\ wenzelm@23252: \ add a (mul r c) \ add b (mul r d)" wenzelm@23252: proof(clarify) wenzelm@23252: assume nz: "r\ r0" and cnd: "c\d" wenzelm@23252: and eq: "add b (mul r c) = add b (mul r d)" wenzelm@23252: hence "mul r c = mul r d" using cnd add_cancel by simp wenzelm@23252: hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)" wenzelm@23252: using mul_0 add_cancel by simp wenzelm@23252: thus "False" using add_mul_solve nz cnd by simp wenzelm@23252: qed wenzelm@23252: chaieb@25250: lemma add_r0_iff: " x = add x a \ a = r0" chaieb@25250: proof- chaieb@25250: have "a = r0 \ add x a = add x r0" by (simp add: add_cancel) chaieb@25250: thus "x = add x a \ a = r0" by (auto simp add: add_c add_0) chaieb@25250: qed chaieb@25250: wenzelm@26462: declare gb_semiring_axioms' [normalizer del] wenzelm@23252: wenzelm@26462: lemmas semiringb_axioms' = semiringb_axioms [normalizer wenzelm@23252: semiring ops: semiring_ops wenzelm@23252: semiring rules: semiring_rules wenzelm@26314: idom rules: noteq_reduce add_scale_eq_noteq] wenzelm@23252: wenzelm@23252: end wenzelm@23252: chaieb@25250: locale ringb = semiringb + gb_ring + chaieb@25250: assumes subr0_iff: "sub x y = r0 \ x = y" wenzelm@23252: begin wenzelm@23252: wenzelm@26462: declare gb_ring_axioms' [normalizer del] wenzelm@23252: wenzelm@26462: lemmas ringb_axioms' = ringb_axioms [normalizer wenzelm@23252: semiring ops: semiring_ops wenzelm@23252: semiring rules: semiring_rules wenzelm@23252: ring ops: ring_ops wenzelm@23252: ring rules: ring_rules chaieb@25250: idom rules: noteq_reduce add_scale_eq_noteq wenzelm@26314: ideal rules: subr0_iff add_r0_iff] wenzelm@23252: wenzelm@23252: end wenzelm@23252: chaieb@25250: wenzelm@23252: lemma no_zero_divirors_neq0: wenzelm@23252: assumes az: "(a::'a::no_zero_divisors) \ 0" wenzelm@23252: and ab: "a*b = 0" shows "b = 0" wenzelm@23252: proof - wenzelm@23252: { assume bz: "b \ 0" wenzelm@23252: from no_zero_divisors [OF az bz] ab have False by blast } wenzelm@23252: thus "b = 0" by blast wenzelm@23252: qed wenzelm@23252: wenzelm@23252: interpretation class_ringb: ringb wenzelm@23252: ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"] nipkow@23477: proof(unfold_locales, simp add: ring_simps power_Suc, auto) wenzelm@23252: fix w x y z ::"'a::{idom,recpower,number_ring}" wenzelm@23252: assume p: "w * y + x * z = w * z + x * y" and ynz: "y \ z" wenzelm@23252: hence ynz': "y - z \ 0" by simp wenzelm@23252: from p have "w * y + x* z - w*z - x*y = 0" by simp nipkow@23477: hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps) nipkow@23477: hence "(y - z) * (w - x) = 0" by (simp add: ring_simps) wenzelm@23252: with no_zero_divirors_neq0 [OF ynz'] wenzelm@23252: have "w - x = 0" by blast wenzelm@23252: thus "w = x" by simp wenzelm@23252: qed wenzelm@23252: wenzelm@26462: declaration {* normalizer_funs @{thm class_ringb.ringb_axioms'} *} wenzelm@23252: wenzelm@23252: interpretation natgb: semiringb wenzelm@23252: ["op +" "op *" "op ^" "0::nat" "1"] nipkow@23477: proof (unfold_locales, simp add: ring_simps power_Suc) wenzelm@23252: fix w x y z ::"nat" wenzelm@23252: { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \ z" wenzelm@23252: hence "y < z \ y > z" by arith wenzelm@23252: moreover { wenzelm@23252: assume lt:"y k. z = y + k \ k > 0" by (rule_tac x="z - y" in exI, auto) wenzelm@23252: then obtain k where kp: "k>0" and yz:"z = y + k" by blast nipkow@23477: from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps) wenzelm@23252: hence "x*k = w*k" by simp wenzelm@23252: hence "w = x" using kp by (simp add: mult_cancel2) } wenzelm@23252: moreover { wenzelm@23252: assume lt: "y >z" hence "\k. y = z + k \ k>0" by (rule_tac x="y - z" in exI, auto) wenzelm@23252: then obtain k where kp: "k>0" and yz:"y = z + k" by blast nipkow@23477: from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps) wenzelm@23252: hence "w*k = x*k" by simp wenzelm@23252: hence "w = x" using kp by (simp add: mult_cancel2)} wenzelm@23252: ultimately have "w=x" by blast } wenzelm@23252: thus "(w * y + x * z = w * z + x * y) = (w = x \ y = z)" by auto wenzelm@23252: qed wenzelm@23252: wenzelm@26462: declaration {* normalizer_funs @{thm natgb.semiringb_axioms'} *} wenzelm@23252: chaieb@23327: locale fieldgb = ringb + gb_field chaieb@23327: begin chaieb@23327: wenzelm@26462: declare gb_field_axioms' [normalizer del] chaieb@23327: wenzelm@26462: lemmas fieldgb_axioms' = fieldgb_axioms [normalizer chaieb@23327: semiring ops: semiring_ops chaieb@23327: semiring rules: semiring_rules chaieb@23327: ring ops: ring_ops chaieb@23327: ring rules: ring_rules chaieb@25250: idom rules: noteq_reduce add_scale_eq_noteq wenzelm@26314: ideal rules: subr0_iff add_r0_iff] wenzelm@26314: chaieb@23327: end chaieb@23327: chaieb@23327: wenzelm@23258: lemmas bool_simps = simp_thms(1-34) wenzelm@23252: lemma dnf: wenzelm@23252: "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))" wenzelm@23252: "(P \ Q) = (Q \ P)" "(P \ Q) = (Q \ P)" wenzelm@23252: by blast+ wenzelm@23252: wenzelm@23252: lemmas weak_dnf_simps = dnf bool_simps wenzelm@23252: wenzelm@23252: lemma nnf_simps: wenzelm@23252: "(\(P \ Q)) = (\P \ \Q)" "(\(P \ Q)) = (\P \ \Q)" "(P \ Q) = (\P \ Q)" wenzelm@23252: "(P = Q) = ((P \ Q) \ (\P \ \ Q))" "(\ \(P)) = P" wenzelm@23252: by blast+ wenzelm@23252: wenzelm@23252: lemma PFalse: wenzelm@23252: "P \ False \ \ P" wenzelm@23252: "\ P \ (P \ False)" wenzelm@23252: by auto wenzelm@23252: use "Tools/Groebner_Basis/groebner.ML" wenzelm@23252: chaieb@23332: method_setup algebra = wenzelm@23458: {* chaieb@23332: let chaieb@23332: fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () chaieb@23332: val addN = "add" chaieb@23332: val delN = "del" chaieb@23332: val any_keyword = keyword addN || keyword delN chaieb@23332: val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; chaieb@23332: in chaieb@23332: fn src => Method.syntax chaieb@23332: ((Scan.optional (keyword addN |-- thms) []) -- chaieb@23332: (Scan.optional (keyword delN |-- thms) [])) src chaieb@23332: #> (fn ((add_ths, del_ths), ctxt) => chaieb@25250: Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt)) chaieb@23332: end chaieb@25250: *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" chaieb@27666: declare dvd_def[algebra] chaieb@27666: declare dvd_eq_mod_eq_0[symmetric, algebra] chaieb@27666: declare nat_mod_div_trivial[algebra] chaieb@27666: declare nat_mod_mod_trivial[algebra] chaieb@27666: declare conjunct1[OF DIVISION_BY_ZERO, algebra] chaieb@27666: declare conjunct2[OF DIVISION_BY_ZERO, algebra] chaieb@27666: declare zmod_zdiv_equality[symmetric,algebra] chaieb@27666: declare zdiv_zmod_equality[symmetric, algebra] chaieb@27666: declare zdiv_zminus_zminus[algebra] chaieb@27666: declare zmod_zminus_zminus[algebra] chaieb@27666: declare zdiv_zminus2[algebra] chaieb@27666: declare zmod_zminus2[algebra] chaieb@27666: declare zdiv_zero[algebra] chaieb@27666: declare zmod_zero[algebra] chaieb@27666: declare zmod_1[algebra] chaieb@27666: declare zdiv_1[algebra] chaieb@27666: declare zmod_minus1_right[algebra] chaieb@27666: declare zdiv_minus1_right[algebra] chaieb@27666: declare mod_div_trivial[algebra] chaieb@27666: declare mod_mod_trivial[algebra] chaieb@27666: declare zmod_zmult_self1[algebra] chaieb@27666: declare zmod_zmult_self2[algebra] chaieb@27666: declare zmod_eq_0_iff[algebra] chaieb@27666: declare zdvd_0_left[algebra] chaieb@27666: declare zdvd1_eq[algebra] chaieb@27666: declare zmod_eq_dvd_iff[algebra] chaieb@27666: declare nat_mod_eq_iff[algebra] wenzelm@23252: haftmann@28402: haftmann@28402: subsection{* Groebner Bases for fields *} haftmann@28402: haftmann@28402: interpretation class_fieldgb: haftmann@28402: fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse) haftmann@28402: haftmann@28402: lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp haftmann@28402: lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0" haftmann@28402: by simp haftmann@28402: lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)" haftmann@28402: by simp haftmann@28402: lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" haftmann@28402: by simp haftmann@28402: lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" haftmann@28402: by simp haftmann@28402: haftmann@28402: lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp haftmann@28402: haftmann@28402: lemma add_frac_num: "y\ 0 \ (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y" haftmann@28402: by (simp add: add_divide_distrib) haftmann@28402: lemma add_num_frac: "y\ 0 \ z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y" haftmann@28402: by (simp add: add_divide_distrib) haftmann@28402: haftmann@28402: haftmann@28402: ML{* haftmann@28402: local haftmann@28402: val zr = @{cpat "0"} haftmann@28402: val zT = ctyp_of_term zr haftmann@28402: val geq = @{cpat "op ="} haftmann@28402: val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd haftmann@28402: val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} haftmann@28402: val add_frac_num = mk_meta_eq @{thm "add_frac_num"} haftmann@28402: val add_num_frac = mk_meta_eq @{thm "add_num_frac"} haftmann@28402: haftmann@28402: fun prove_nz ss T t = haftmann@28402: let haftmann@28402: val z = instantiate_cterm ([(zT,T)],[]) zr haftmann@28402: val eq = instantiate_cterm ([(eqT,T)],[]) geq haftmann@28402: val th = Simplifier.rewrite (ss addsimps simp_thms) haftmann@28402: (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} haftmann@28402: (Thm.capply (Thm.capply eq t) z))) haftmann@28402: in equal_elim (symmetric th) TrueI haftmann@28402: end haftmann@28402: haftmann@28402: fun proc phi ss ct = haftmann@28402: let haftmann@28402: val ((x,y),(w,z)) = haftmann@28402: (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct haftmann@28402: val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] haftmann@28402: val T = ctyp_of_term x haftmann@28402: val [y_nz, z_nz] = map (prove_nz ss T) [y, z] haftmann@28402: val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq haftmann@28402: in SOME (implies_elim (implies_elim th y_nz) z_nz) haftmann@28402: end haftmann@28402: handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE haftmann@28402: haftmann@28402: fun proc2 phi ss ct = haftmann@28402: let haftmann@28402: val (l,r) = Thm.dest_binop ct haftmann@28402: val T = ctyp_of_term l haftmann@28402: in (case (term_of l, term_of r) of haftmann@28402: (Const(@{const_name "HOL.divide"},_)$_$_, _) => haftmann@28402: let val (x,y) = Thm.dest_binop l val z = r haftmann@28402: val _ = map (HOLogic.dest_number o term_of) [x,y,z] haftmann@28402: val ynz = prove_nz ss T y haftmann@28402: in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) haftmann@28402: end haftmann@28402: | (_, Const (@{const_name "HOL.divide"},_)$_$_) => haftmann@28402: let val (x,y) = Thm.dest_binop r val z = l haftmann@28402: val _ = map (HOLogic.dest_number o term_of) [x,y,z] haftmann@28402: val ynz = prove_nz ss T y haftmann@28402: in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) haftmann@28402: end haftmann@28402: | _ => NONE) haftmann@28402: end haftmann@28402: handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE haftmann@28402: haftmann@28402: fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b haftmann@28402: | is_number t = can HOLogic.dest_number t haftmann@28402: haftmann@28402: val is_number = is_number o term_of haftmann@28402: haftmann@28402: fun proc3 phi ss ct = haftmann@28402: (case term_of ct of haftmann@28402: Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => haftmann@28402: let haftmann@28402: val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => haftmann@28402: let haftmann@28402: val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => haftmann@28402: let haftmann@28402: val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => haftmann@28402: let haftmann@28402: val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => haftmann@28402: let haftmann@28402: val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => haftmann@28402: let haftmann@28402: val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop haftmann@28402: val _ = map is_number [a,b,c] haftmann@28402: val T = ctyp_of_term c haftmann@28402: val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} haftmann@28402: in SOME (mk_meta_eq th) end haftmann@28402: | _ => NONE) haftmann@28402: handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE haftmann@28402: haftmann@28402: val add_frac_frac_simproc = haftmann@28402: make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], haftmann@28402: name = "add_frac_frac_simproc", haftmann@28402: proc = proc, identifier = []} haftmann@28402: haftmann@28402: val add_frac_num_simproc = haftmann@28402: make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], haftmann@28402: name = "add_frac_num_simproc", haftmann@28402: proc = proc2, identifier = []} haftmann@28402: haftmann@28402: val ord_frac_simproc = haftmann@28402: make_simproc haftmann@28402: {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, haftmann@28402: @{cpat "(?a::(?'a::{field, ord}))/?b \ ?c"}, haftmann@28402: @{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, haftmann@28402: @{cpat "?c \ (?a::(?'a::{field, ord}))/?b"}, haftmann@28402: @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, haftmann@28402: @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], haftmann@28402: name = "ord_frac_simproc", proc = proc3, identifier = []} haftmann@28402: haftmann@28402: val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", haftmann@28402: "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"] haftmann@28402: haftmann@28402: val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", haftmann@28402: "add_Suc", "add_number_of_left", "mult_number_of_left", haftmann@28402: "Suc_eq_add_numeral_1"])@ haftmann@28402: (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) haftmann@28402: @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} haftmann@28402: val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, haftmann@28402: @{thm "divide_Numeral1"}, haftmann@28402: @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"}, haftmann@28402: @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, haftmann@28402: @{thm "mult_num_frac"}, @{thm "mult_frac_num"}, haftmann@28402: @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, haftmann@28402: @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, haftmann@28402: @{thm "diff_def"}, @{thm "minus_divide_left"}, haftmann@28402: @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym] haftmann@28402: haftmann@28402: local haftmann@28402: open Conv haftmann@28402: in haftmann@28402: val comp_conv = (Simplifier.rewrite haftmann@28402: (HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"} haftmann@28402: addsimps ths addsimps comp_arith addsimps simp_thms haftmann@28402: addsimprocs field_cancel_numeral_factors haftmann@28402: addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, haftmann@28402: ord_frac_simproc] haftmann@28402: addcongs [@{thm "if_weak_cong"}])) haftmann@28402: then_conv (Simplifier.rewrite (HOL_basic_ss addsimps haftmann@28402: [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)})) wenzelm@23252: end haftmann@28402: haftmann@28402: fun numeral_is_const ct = haftmann@28402: case term_of ct of haftmann@28402: Const (@{const_name "HOL.divide"},_) $ a $ b => haftmann@28402: numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct) haftmann@28402: | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct) haftmann@28402: | t => can HOLogic.dest_number t haftmann@28402: haftmann@28402: fun dest_const ct = ((case term_of ct of haftmann@28402: Const (@{const_name "HOL.divide"},_) $ a $ b=> haftmann@28402: Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) haftmann@28402: | t => Rat.rat_of_int (snd (HOLogic.dest_number t))) haftmann@28402: handle TERM _ => error "ring_dest_const") haftmann@28402: haftmann@28402: fun mk_const phi cT x = haftmann@28402: let val (a, b) = Rat.quotient_of_rat x haftmann@28402: in if b = 1 then Numeral.mk_cnumber cT a haftmann@28402: else Thm.capply haftmann@28402: (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) haftmann@28402: (Numeral.mk_cnumber cT a)) haftmann@28402: (Numeral.mk_cnumber cT b) haftmann@28402: end haftmann@28402: haftmann@28402: in haftmann@28402: val field_comp_conv = comp_conv; haftmann@28402: val fieldgb_declaration = haftmann@28402: NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'} haftmann@28402: {is_const = K numeral_is_const, haftmann@28402: dest_const = K dest_const, haftmann@28402: mk_const = mk_const, haftmann@28402: conv = K (K comp_conv)} haftmann@28402: end; haftmann@28402: *} haftmann@28402: haftmann@28402: declaration fieldgb_declaration haftmann@28402: haftmann@28402: end